Matrix Rank Calculator

The rank of a matrix is the number of linearly independent rows — or equivalently, the number of linearly independent columns. They always match. Rank tells you the dimension of the column space: how many distinct directions the matrix can produce in its output.

A full-rank square matrix (rank = n for an n x n matrix) is invertible. A rank-deficient matrix is singular — it collapses some inputs to zero output.

For a system of equations Ax = b, rank determines solvability. If rank(A) equals rank of the augmented matrix [A|b] and both equal n (the number of unknowns), there is exactly one solution. If rank(A) = rank([A|b]) but both are less than n, there are infinitely many solutions. If rank(A) is less than rank([A|b]), there is no solution.

Nullity = n - rank. It counts the dimensions of the null space — the set of all vectors x such that Ax = 0. The rank-nullity theorem guarantees that rank + nullity always equals the number of columns.

This calculator uses Gaussian elimination with partial pivoting. Each step: find the largest absolute value in the current column at or below the current pivot row, swap it to the pivot position, scale to make the pivot 1, then eliminate that column in all other rows.

Enter format: “1,2,3;4,5,6;7,8,9” for a 3x3 matrix. Semicolons separate rows.

Example: the 3x3 matrix with rows [1,2,3], [4,5,6], [7,8,9] has rank 2, not 3 — the third row is the second row minus the first (up to scaling), so the rows are not all independent. Nullity = 3 - 2 = 1.